Problem: Solve for $x$ : $ 2|x - 10| + 8 = 4|x - 10| + 2 $
Explanation: Subtract $ {2|x - 10|} $ from both sides: $ \begin{eqnarray} 2|x - 10| + 8 &=& 4|x - 10| + 2 \\ \\ {- 2|x - 10|} && {- 2|x - 10|} \\ \\ 8 &=& 2|x - 10| + 2 \end{eqnarray} $ Subtract $2$ from both sides: $ \begin{eqnarray} 8 &=& 2|x - 10| + 2 \\ \\ {- 2} && {- 2} \\ \\ 6 &=& 2|x - 10| \end{eqnarray} $ Divide both sides by ${2}$ $ \dfrac{6} {{2}} = \dfrac{2|x - 10|} {{2}} $ Simplify: $ 3 = |x - 10| $ Because the absolute value of an expression is its distance from zero, it has two solutions, one negative and one positive: $ -3 = x - 10 $ or $ 3 = x - 10 $ Solve for the solution where $x - 10$ is negative: $ - 3 = x - 10$ Add ${10}$ to both sides: $ \begin{eqnarray} - 3 &=& x - 10 \\ \\ {+ 10} && {+ 10} \\ \\ -3 + 10 &=& x \end{eqnarray} $ $ 7 = x $ Then calculate the solution where $x - 10$ is positive: $ 3 = x - 10 $ Add ${10}$ to both sides: $ \begin{eqnarray} 3 &=& x - 10 \\ \\ {+ 10} && {+ 10} \\ \\ 3 + 10 &=& x \end{eqnarray} $ $ 13 = x $ Thus, the correct answer is $x = 7 $ or $x = 13 $.